15 problems

Let $M^n$ be a closed, minimally immersed submanifold in the unit sphere $\mathbb{S}^{n+m}$ with constant scalar curvature, equivalently with constant length of the second fundamen…

Let $M$ be a biharmonic submanifold of a sphere. BMO conjecture. The submanifold $M$ has constant mean curvature. The source presents this as one of the well-known open conjectures…

Balmus–Montaldo–Oniciuc constant-mean-curvature conjecture. Any proper biharmonic submanifold in $\mathbb{S}^{m+1}$ is CMC.

Let $S^{d-1}$ be a unit sphere in $\mathbb R^d$. Suppose we are given $d$ points $x_1, \ldots, x_d\in S^{d-1}$ and a continuous function $f: S^{d-1}\to \mathbb R$. Knaster's conjec…

Let $T\subset S^3$ be an embedded minimal torus, and let $T_{\mathrm{Cl}}$ denote the Clifford torus in $S^3$. Lawson's conjecture. There exists an isometry $F$ of $S^3$ such that…

Let $M$ be a non-minimal biharmonic submanifold of the unit sphere $S^n(1)$. Constant-mean-curvature conjecture. The mean curvature of $M$ is constant. The claim is a weaker statem…

Classification conjecture. The only proper biharmonic hypersurfaces in $\mathbb{S}^{m+1}$ are the open parts of hyperspheres $\mathbb{S}^m(1/\sqrt{2})$ or of the standard products…

Let $\mathcal{I}_7$ denote the class of seven-vertex polytopes inscribed in the unit sphere $\mathbb{S}^2$, and let $S(P)$ be the surface area of $P$. Let $e_3$ be the third standa…

Let $M^n$ be a closed, minimally immersed hypersurface of the unit sphere $\mathbb{S}^{n+1}$ with constant scalar curvature. Stronger version of Chern's conjecture. Then $M^n$ is i…

Chern's conjecture. The value of $|B|^2$ must lie in a discrete subset of $\mathbb{R}$.

Let $S^2=\{u\in\mathbb{R}^3:|u|=1\}$ be the 2-dimensional sphere, and for $n\geq 2$ let $\mu_n(S^2)$ denote the maximal number of $3$-term arithmetic progressions in an $n$-element…

Horizontal diameter rigidity conjecture. For any singular Riemannian foliation on a unit sphere $\mathbb{S}^{n}$, we have

Maehara's conjecture. All the spheres in $\mathfrak{F}$ have a point in common.

The weak Knaster conjecture. There exists $n=n(l)$ such that, for every such $X$ and $f$, there is a rotation $\rho\in O(n)$ satisfying

Let $S^2$ be a unit sphere in $\mathbb R^3$. Suppose we are given $4$ points $x_1, \ldots, x_4\in S^2$ lying on a single circle, and a continuous function $f:S^2\to\mathbb R$. Make…